963.00 First Power: One Dimension 
963.01 In conventional XYZ coordination, onedimensionality is identified geometrically with linear pointal frequency. The linear measure is the first power, or the edge of the square face of a cube. 
963.02 In synergetics, the firstpower linear measure is the radius of the sphere. 
Table 963.10 
963.10 Synergetics Constant: The synergetics constant was evolved to convert thirdpower, volumetric evaluation from a cubical to a tetrahedral base and to employ the ABCDfourdimensional system's vector as the linear computational input. In the case of the cube this is the diagonal of the cube's square face. Other power values are shown in Table 963.10. We have to find the total vector powers involved in the calculation. In synergetics we are always dealing in energy content: when vector edges double together in quadrivalence or octavalence, the energy content doubles and fourfolds, respectively. When the vector edges are halfdoubled together, as in the icosahedron phase of the jitterbug^{__}halfway between the vector equilibrium 20 and the octahedron compression^{__}to fourfold and fivefold contraction with the vectors only doubled, we can understand that the volume of energy in the icosahedron (which is probably the same 20 as that of the vector equilibrium) is just compressed. (See Secs. 982.45 and 982.54.) 
964.00 Second Power: Two Dimensions 
964.01 In conventional XYZ coordination, twodimensionality is identified with areal pointal frequency. 
964.02 In synergetics, second powering = point aggregate quanta = area. In synergetics, second powering represents the rate of system surface growth. 
964.10 Spherical Growth Rate: In a radiational or gravitational wave system, second powering is identified with the point population of the concentrically embracing arrays of any given radius, stated in terms of frequency of modular subdivisions of either the radial or chordal circumference of the system. (From Synergetics Corollary, see Sec. 240.44.) 
965.00 Third Power: Three Dimensions 
965.04 Perpendicularity (90degreeness) uniquely characterizes the limit of three dimensionality. Equiangularity (60degreeness) uniquely characterizes the limits of four dimensional systems. 
966.00 Fourth Power: Four Dimensions 
966.01 In a radiational or gravitational wave system, fourth powering is identified with the interpointal domain volumes. 
966.04 The vectorequilibrium model displays fourdimensional hexagonal central cross section. 
Fig. 966.05 
966.05 Arithmetical fourthpower energy evolution order has been manifest time and again in experimental physics, but could not be modelably accommodated by the XYZ c.g_{t}.s. system. That the fourth dimension can be modelably accommodated by synergetics is the result of complex local intertransformabilities because the vector equilibrium has, at initial frequency zero, an inherent volume of 20. Only eight cubes can be closest packed in omnidirectional embracement of any one point in the XYZ system: in the third powering of two, which is eight, all pointsurrounding space has been occupied. In synergetics, third powering is allspacefillingly accounted in tetrahedral volume increments; 20 unit volume tetrahedra closepack around one point, which point surrounding reoccurs isotropically in the centers of the vector equilibria. When the volume around one is 20, the frequency of the system is at one. When the XYZ system modular frequency is at one, the cube volume is one, while in the vectorequilibrium synergetic system, the initial volume is 20. When the frequency of modular subdivision of XYZ cubes reads two, the volume is eight. When the vector equilibria's module reads two, the volume is 20F^{3} = 20 × 8 = 160 tetrahedral volumes^{__}160 = 25 × 5^{__}thus demonstrating the use of conceptual models for fourth and fifthpowering volumetric growth rates. With the initial frequency of one and the volume of the vector equilibrium at 20, it also has 24 × 20 A and B Quanta Modules; ergo is inherently initially 480 quanta modules. 480 = 25 × 5 × 3. With frequency of two the vector equilibrium is 160 × 24 = 3840 quanta modules. 3840 = 28 × 3 × 5. (See Illus. 966.05.) 
966.07 In an omnimotional Universe, it is possible to join or lock together two previously independently moving parts of the system without immobilizing the remainder of the system, because fourdimensionality allows local fixities without in any way locking or blocking the rest of the system's omnimotioning or intertransforming. This independence of local formulation corresponds exactly with life experiences in Universe. This omnifreedom is calculatively accommodated by synergetics' fourth and fifthpower transformabilities. (See Sec. 465, "Rotation of Wheels or Cams in Vector Equilibrium.") (See Illus. 465.01.) 
970.00 First and ThirdPower Progressions of Vector Equilibria
970.10 Rationality of Planar Domains and Interstices: There is a 12F^{2} + 2 omniplanarbound, volumetricdomain marriage with the 10F^{2} + 2 strictly spherical shell accounting. (See tables at Sec. 955.40 and at Sec. 971.00.) 
Fig. 970.20 
970.20 Spheres and Spaces: The successive (20F^{3})  20 (F  1)^{3} layershell, planarbound, tetrahedral volumes embrace only the tangential inner and outer portions of the concentrically closestpacked spheres, each of whose respective complete concentric shell layers always number 10F^{2} + 2. The volume of each concentric vectorequilibrium layer is defined and structured by the isotropic vector matrix, or octet truss, occurring between the spherical centers of any two concentricsphere layers of the vector equilibrium, the inner part of one sphere layer and the outer part of the other, with only the center or nuclear ball being both its inner and outer parts. 
970.21 There is realized herewith a philosophical synergetic sublimity of omnirational, universal, holistic, geometrical accounting of spheres and spaces without recourse to the transcendentally irrational pi . (See drawings section.) (See Secs. 954.56 and 1032.) 
971.00 Table of Basic Vector Equilibrium Shell Volumes 
Fig. 971.01 
971.01 Relationships Between First and Third Powers of F Correlated to ClosestPacked Triangular Number Progression and ClosestPacked Tetrahedral Number Progression, Modified Both Additively and Multiplicatively in Whole Rhythmically Occurring Increments of Zero, One, Two, Three, Four, Five, Six, Ten, and Twelve, All as Related to the Arithmetical and Geometrical Progressions, Respectively, of Triangularly and Tetrahedrally ClosestPacked Sphere Numbers and Their Successive Respective Volumetric Domains, All Correlated with the Respective Sphere Numbers and Overall Volumetric Domains of Progressively Embracing Concentric Shells of Vector Equilibria: Short Title: Concentric Sphere Shell Growth Rates. 
971.02 The red zigzag between Columns 2 and 3 shows the progressive, additive, triangularsphere layers accumulating progressively to produce the regular tetrahedra. 
971.04 Column 5 is the tetrahedral number with the new nucleus. 
971.05 In Column 6, the integer SIX functions as zero in the same manner in which NINE functions innocuously as zero in all arithmetical operations. 
971.06 In Column 6, we multiply Column 5 by a constant SIX, to the product of which we add the sixstage 0, 1, 2, 3, 4, 5 wavefactor growth crest and break of Column 7. 
971.09
Column 10 lists the cumulative, planarbound, tetrahedral
volumes of the
arithmetical progression of third powers of the successive
frequencies of whole vector
equilibria. The vector equilibrium's initial nonfrequencied
tetravolume, i.e., its quantum
value, is 20. The formula for obtaining the frequencyprogressed
volumes of vector
equilibrium is:
Volume of VE = 20F^{3}.

971.10
In Column 11, we subtract the previous frequencyvector
equilibrium's
cumulative volume from the new onefrequencygreater
vector equilibrium's cumulative
volume, which yields the tetrahedral volume of the outermost
shell. The outer vector
equilibrium's volume is found always to be:

971.11
Incidentally, the

Next Section: 971.20 